Problem Statement
The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of n points (data pairs), i = 1, ..., n, where is an independent variable and is a dependent variable whose value is found by observation. The model function has the form, where the m adjustable parameters are held in the vector . The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, S, of squared residuals
is a minimum. A residual is defined as the difference between the actual value of the dependent variable and the value predicted by the model.
- .
An example of a model is that of the straight line. Denoting the intercept as and the slope as, the model function is given by . See linear least squares for a fully worked out example of this model.
A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, x and z, say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point.
Read more about this topic: Least Squares
Famous quotes containing the words problem and/or statement:
“The problem of culture is seldom grasped correctly. The goal of a culture is not the greatest possible happiness of a people, nor is it the unhindered development of all their talents; instead, culture shows itself in the correct proportion of these developments. Its aim points beyond earthly happiness: the production of great works is the aim of culture.”
—Friedrich Nietzsche (1844–1900)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)